If I had photographs of piles of leaves from different forest types – and some of the piles in different photographs were from the same forest type – and I was only able to identify the leaves by geometry and color and not by their name – and the leaves had deteriorated so much that I couldn’t use any traditional geometrical classification to identify them by their shape – though they all had a unique shape, what type of math could I use to group the photographs by forest types? What if the leaves pretty much all had the same shape – though they were known to be different and from different forest types, and the only information that I could use to identify the leaves was a video that showed that different leaves changed their shape at different rates while they were deteriorating.

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She 'tis not there.

 

Actually, the problem could apply to boundaries within the same forest.

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She 'tis not there.

 

Is this for a botany class? Your question is somewhat confusing. Can you word your homework question exactly as it was written?

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Pay it forward.

 

It's not really a homework question. I was wondering if there was a way of characterizing different piles of leaves and how they deteriorate geometrically without nomenclature that isn't mathematical - if math has a nomenclature. I imagine that there are different ways of recognizing patterns in forests - the distribution of different diameter trees having bark with different patterns, how branch patterns change as a function of depth, how these things overlap at different angles in a forest, the distribution of crown characteristics as viewed from above, etc. Develop this idea too much and a person might put a lot of timber cruisers out of work. I was interested in the piles. I don't know a lot about math. I've learned some things about curves and surfaces. The leaf pile problem seems as though it would involve an analysis of different types of distributions. The leafs might loose all of the characteristics that are used to identified them by people who are interested in forests if they are laying on their side, covered up by other leaves, and only information about their sides revealed from an image-capture system were available. I had asked a question in the first post to this thread about characterizing the piles by how they deteriorate given - hypothetically, that all the leaves in the piles changed their shape from more or less the same form to the same form - but different types of leaves changed at different rates. Solving this problem might have to involve an analysis of how the pile collapses. This might be a little too much for me to think about at this time. I'm curious, but I'd mostly like to know about a type of math that could be used to characterize the piles having leaves with different geometric characteristics at this time without a consideration of how they change shape as they deteriorate. I’d like to suppose that no information is available about how the piles were formed as different types of leaves fell and formed piles - or areas where there was a larger depth of leaves that may be considered piles.

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She 'tis not there.

 

Maybe you should utilise a ven digram with shape,location size, etc. to derive a basic hypothosis. I always tuaght my students to go back to nuts and bolts before jumping in. Time is an iomportant factor here.

Think outside the square then square it.

 

Maybe periods of the frequency of the presence of different leaf types could be represented with Venn diagrams - along with minimum numbers of types of leaves and locations along an x axis. Is there a typical way of using Venn diagrams with this type or problem?

I've been thinking about how to represent arrangement and it is really puzzling. I've thought about identifying locations in the pile and then for each location identifying the frequency that each type of leave appears from zero to 360 degrees. Or, identifying the distance between locations where this frequency representation changes significantly. It might be useful to represent the location of the leaves to each other. However, if I tried to represent this by adding frequency representations of different leaves, what I might wind up with is a a sum and not a ratio. A ratio wouldn't represent the number of leaves - or it might exaggerate it. Maybe if I plotted the sum of the frequency representation on one axis and the ratio of the frequency representations on another axis - and did this for a whole bunch of different leaf shapes, I might come up with something representative. However, how could I define leaves with similar shapes and the shapes plotted on the two axes? Can topology and statistics be used for this? What if the information on the two axes wasn't a function? What I think might be really helpful is if someone referred me to a book that described the use and basics - in the language of laypeople, of the different types of math listed in University catalogs.

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She 'tis not there.

 

As a practical matter, there are two basic problems involved. First is the problem of looking at a photo and interpreting what is in it - i.e. whether a blob is a leaf or a pile of leaves, tracing the outline of a leaf as a curve, etc. If you rely on human beings to do this, it would be very labor intensive. So it would be nice to have computer programs to do the job. However, the mathematics needed to program computers to recognize what is in a picture is not perfected although there are many methods that work in special situations. Efforts to solve the problem are currently topics in Computer Science rather than topics in Mathematics. The general topic is "Pattern Recognition". Specific methods would be "Neural Nets", "Fuzzy Classification", "Attention Selection".

Let's assume you have gotten data about types of leaves or curves that outline leaves. Telling whether two curves are "similar" is a special case of the first problem, which was recognizing what is in a picture. I think I can find Computer Science papers written about this problem, but it is not a separate branch of Mathematics. If you have data that does not represent a shape (such as leaf length, width, area,... etc.) then the mathematics called "Statistical Pattern Recognition" is applicable. (Don't restrict your thinking to an x-y graph in two dimensions.) Statistical Pattern Recognition may not do a job. Whether it works depends on whether your data really has enough information to solve our problem. (The mathematical field called "Statistics" is not the same as "Statistical Pattern Recognition".)

 

Let's say that in addition to identifying the leaves in a pile or mound I also wanted to classify forest areas by the distribution of different leaf types in piles in the same area. What if I also wanted to represent patterns of how this distribution changed in the same classification of forest areas from season to season?

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She 'tis not there.

 

Ok, let's say you do. But are you doing conceptual daydreaming or are you actually going to do something concrete? If you are going to do something concrete then you have to be specific about the data that you have. Can you give an example of the type of data that you are imagining?

 

I'm not sure what you mean. Leaves are concrete. If it would be easier to model something more popular, cyclic molecules of different shapes could be modeled instead of leaves. The time scale in this case would be in fractions of a second during a domino reaction instead of seasons. However, the mixture might have to be heterogeneous because I would like to model where the same types of molecules appear at different times.

Here's another idea: representing the shapes that form after ripples in a pond spreading from raindrops converge to form sequences of different changing shapes that might be represented by many intersecting curves – some curves increasing the area of a polygon that would result from imaginary lines connecting where the curves representing the ripples intersect - and some curves decreasing this area.

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She 'tis not there.

 

Leaves are concrete things, but your descriptions of the data you intend to collect about leaves is not specific. Give a numerical example of the type of data you are talking about if you conceptualize anything specific. Otherwise, you are just daydreaming. It's fine to daydream, but you don't need our help to do that.

 

O.K. Let's call them leaf type a, s, d, and f. Lets say that o represents a hole in the pile. Let's take a large rectangular subdivision of a pile as a sample. An oversimplification might look like this:

sdfasdfaodfasd
sdadoadsfasdfa
asdadfadsoasdf

Next season it may look like this:

sdodsfadsfadsf
sdodsfasdadsfa
sdadsfasdfodsa

Next season:

asdadsfodsadsa
sdoasdfadsfadsf
sdadsfadsadofas

Next:

asdfasdfadsfosd
dsasdadsadfoasf
sdadfsasofasdfa

Next:

sdosdfasdfadsad
sdadsoasdfasdfa
asdadfasdfodsas

What are the names of all patterns that can appear as the samples change and what types of math can be used to represent how they change?

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She 'tis not there.

 

Real leaves overlap and don't form neat layers, so I think you are dealing with an abstract conceptual problem, not a practical one. Before we worry about how many patterns of samples can exist (which is the field of "Combinatorics"), let's not loose sight of the original question, which was something about "forest types". You haven't given any example of what you mean by a "forest type".

 

I'd like to understand this problem on the conceptual level and not the practical level by identifying the patterns that you wrote about if they can be described.

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She 'tis not there.

 

We have to be clear about what "this problem" is. Does it have to do with the idea of estimating the "forrest type" from the leaf piles? We can use your model of leaf piles, if you wish. But if you want to estimate forrest type, we have to define what that means.

If you want count all possible types of leaf piles, that's a different problem. It doesn't necessarily have anything to do with estimating the properties of forrests.